Pl\"unnecke inequalities for measure graphs with applications

TL;DR

This paper extends Pl"unnecke's graph inequality to measure graphs, providing new inequalities for measure-preserving actions and improving density estimates in countable abelian groups.

Contribution

It generalizes Pl"unnecke inequalities to measure graphs and applies them to obtain improved density bounds in abelian groups.

Findings

New Pl"unnecke inequalities for measure graphs

Enhanced Banach density estimates in abelian groups

Applications via Furstenberg correspondence principle

Abstract

We generalize Petridis's new proof of Pl\"unnecke's graph inequality to graphs whose vertex set is a measure space. Consequently, this gives new Pl\"unnecke inequalities for measure preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that improve on those given by Jin.